Illuminance & the Lighting Laws
Half as far means four times as bright; tilt the page and it dims.
Lesson at a glanceShow the key points
- Illuminance E = dF/dA (22.4), luminous flux per unit area; unit lux = lm/m². It sets how bright a surface looks.
- From a point source of intensity I: E = I cos θ / r² (22.5): inverse-square law, times the cosine of the angle between the surface normal and the light.
- Lambert's cosine law: a flat source's intensity at angle θ from its normal is I = I₀ cos θ (22.6); such surfaces are perfectly diffusing.
How brightly lit your page looks depends on how much light lands on each square centimetre, which depends on how far the lamp is and at what slant its light arrives.
Illuminance
The inverse-square and cosine laws
A small area ΔA is at distance r from a point source of intensity I, and its normal makes an angle θ with the line to the source. Seen from the source, the area subtends a solid angle
so it receives flux ΔF = IΔω, and its illuminance is
A small bulb radiating 1260 lm evenly in all directions hangs 0.60 m above a table. Find the illuminance (a) directly below it, (b) at a point on the table 1.0 m from the bulb.
Lambert’s cosine law
A small flat source (a glowing patch, or a sheet of paper lit from behind) radiates only into the half-space in front of it, and not equally in all those directions. Its intensity is greatest along the normal and falls off away from it. For many surfaces:
A perfectly diffusing small source has intensity 200 cd along its normal. How much flux does it send into a cone of 0.015 sr around a direction 45° from the normal?
- 1Illuminance E = dF/dA, in lux (lm/m²).
- 2From a point source: E = I cos θ/r², the inverse-square law with a cosine factor.
- 3Lambert's cosine law for a flat source: I = I₀ cos θ (perfectly diffusing surfaces).
Check yourself
Keep exploring
Hand-picked videos, simulations and readings that explain this idea from a different angle.